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Question
The number of solutions of `("d"y)/("d"x) = (y + 1)/(x - 1)` when y (1) = 2 is ______.
Options
None
One
Two
Infinite
Solution
The number of solutions of `("d"y)/("d"x) = (y + 1)/(x - 1)` when y (1) = 2 is one.
Explanation:
The given differential equation is `("d"y)/("d"x) = (y + 1)/(x - 1)`
⇒ `("d"y)/(y + 1) = ("d"x)/(x - 1)`
Integrating both sides, we get
`int ("d"y)/(y + 1) = int ("d"x)/(x - 1)`
⇒ log(y + 1) = log(x – 1) + log c
⇒ log(y + 1) – log(x – 1) = log c
⇒ `log|(y + 1)/(x - 1)|` = log c
⇒ `(y + 1)/(x - 1)` = c
Put x = 1 and y = 2
⇒ `(2 + 1)/(1 - 1)` = c
∴ c = `oo`
∴ `(y +1)/(x - 1) = 1/0`
⇒ x – 1 = 0
⇒ x = 1.
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